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Properties

Label 2-770-11.4-c1-0-2
Degree $2$
Conductor $770$
Sign $-0.444 + 0.895i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.785 + 2.41i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−2.05 + 1.49i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−2.80 − 2.03i)9-s − 10-s + (−2.31 − 2.37i)11-s − 2.54·12-s + (−5.19 − 3.77i)13-s + (0.309 − 0.951i)14-s + (−0.785 − 2.41i)15-s + (−0.809 + 0.587i)16-s + (−3.19 + 2.32i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.453 + 1.39i)3-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.840 + 0.610i)6-s + (−0.116 − 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.935 − 0.679i)9-s − 0.316·10-s + (−0.698 − 0.716i)11-s − 0.734·12-s + (−1.44 − 1.04i)13-s + (0.0825 − 0.254i)14-s + (−0.202 − 0.624i)15-s + (−0.202 + 0.146i)16-s + (−0.775 + 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257966 - 0.415815i\)
\(L(\frac12)\) \(\approx\) \(0.257966 - 0.415815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.31 + 2.37i)T \)
good3 \( 1 + (0.785 - 2.41i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (5.19 + 3.77i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.19 - 2.32i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.762 - 2.34i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.99T + 23T^{2} \)
29 \( 1 + (-2.79 - 8.60i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.38 + 4.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.41 - 10.5i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.14 + 6.61i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + (2.58 - 7.96i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.14 + 5.91i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.26 + 3.89i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.91 - 2.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + (-4.77 + 3.46i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.16 - 9.75i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.02 + 3.65i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.7 - 7.84i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (-9.67 - 7.03i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.77194143710165034865979618569, −10.30413806261995455622121465960, −9.322158110860574120315218629178, −8.241076554123841442915354532556, −7.41013939258990265513410581571, −6.31995536976242121955918335639, −5.23446651835395171791942252161, −4.79665329787478811301377527847, −3.66972018302296327565319215062, −2.88921773760290214305140748213, 0.19984033280902606974979048565, 1.93548848432849530148972990373, 2.65225783730698662989270318045, 4.47571136280249407088756575802, 5.11641385212134864879735104006, 6.30602019735782191135077472362, 7.13501654448164634095245117916, 7.58543691733275132690647396414, 8.964693466377806670845382582403, 9.743216692410269824890921980188

Graph of the $Z$-function along the critical line